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We study the numerical solution of a Boundary Value problem for second order quadratic differential equations which arises in the numerical prediction of meteorological parameters. In the present work, we use finite differences and focus on the numerical solution of the resulting nonlinear system. More precisely, we apply classical Newton’s and Quasi-Newton methods paying attention to the special sparse form of the Jacobian matrix and modify appropriately the LU factorization in order to reduce significantly the required floating point operations. Furthermore, we implement and study in depth the behavior of all the proposed procedures in respect of their accuracy, stability and complexity, using data from South East Mediterranean Sea. All the methods are tested with a variety of initial values and their performance is presented and discussed leading to interesting results on the sensitivity of the selected starting point.

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